However, if you want to write data, you need to send a transaction which means you need a private key (basically a wallet).

There are a few Ethereum Mobile browsers on the market like Status, Toshi or Cipher. I just checked and no one accepts yet to connect to a private node via RPC (only public networks like mainet, ropsten, rinkeby). You will need to wait the feature to be implemented on one of those or a new Ethereum mobile wallet with this feature.